Anti-periodic solutions to a class of non-monotone evolution equations (Q1576815)

The authors follow the ideas of H. Okochi on noncoercitive evolution equations with antiperiodic boundary conditions. The existence of solutions to the problem \[ au_t(t,x) +Au(t,x) -b u(t,x) +f(x, u(t,x))\ni h(t,x), \quad t\in[0,T], x\in\Omega, \] \[ u(0,x)=-u(T,x), \] with \(h\in L^2(0,T; L_2(\Omega))\) is considered. Under the assumption that \(A\) is a maximal monotone operator in \(L^2(\Omega)\) and a subdifferential (in \(L^2(\Omega)\)) of a functional \(\varphi: L^2(\Omega)\to (-\infty, \infty], \;\) and some continuity assumption on the Carathéodory function \( f: \Omega\times {\mathbb{R}}\to {\mathbb{R}}\) it is proved that the problem has at least one generalized solution \(u\in W^{1,2}(0,T; L^2(\Omega))\). A combination of monotonicity and compactness methods is used. An example illustrating the abstract theory is discussed.